Opuscula Math. 31, no. 3 (), 411-424
http://dx.doi.org/10.7494/OpMath.2011.31.3.411
Opuscula Mathematica

# Existence and asymptotic behavior of solutions for Hénon type equations

Abstract. This paper is concerned with ground state solutions for the Hénon type equation $$-\Delta u(x)=|y|^{\alpha} u^{p-1}(x)$$ in $$\Omega$$, where $$\Omega=B^k(0,1)\times B^{n-k}(0,1)\subset \mathbb{R}^n$$ and $$x=(y,z) \in \mathbb{R}^k \times \mathbb{R}^{n-k}$$. We study the existence of cylindrically symmetric and non-cylindrically symmetric ground state solutions for the problem. We also investigate asymptotic behavior of the ground state solution when $$p$$ tends to the critical exponent $$2^*=\frac {2n}{n-2}$$ if $$n\geq 3$$.
Keywords: Hénon equation, cylindrical symmetry, non-cylindrical symmetry, asymptotic behavior.
Mathematics Subject Classification: 35J50, 35J55, 35J60.